WIAS Preprint No. 1500, (2010)

On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Zelik, Sergey

2010 Mathematics Subject Classification

  • 35K55 34E15

Keywords

  • Doubly nonlinear equations, parametrized solutions, rate-independent systems, vanishing-viscosity limit, jump path

Abstract

We consider quasilinear parabolic systems with a nonsmooth rate-independent dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity $varepsilon>0$. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parametrization, where the norm associated with the viscosity is used to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allows us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems.

Appeared in

  • Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XIII (2014) pp. 67--135.

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